§ 5

Vectors are constructed perpendicular to the faces of an arbitrary tetrahedron, equal in length to the areas of these faces and directed towards the vertices opposite to the faces. Prove that the sum of these vectors is equal to 0.

A trihedral angle is given. Using the properties of the cross product, find an expression for one of its dihedral angles in terms of the plane angles.

Suppose that in an oriented plane a positive basis is given such that $\left|\mathbf{e}_{1}\right|=2,\left|\mathbf{e}_{2}\right|=3$ and $\left(\mathbf{e}_{1}, \mathbf{e}_{2}\right)=2$. Find the area of the oriented parallelogram built on the vectors $\mathbf{a}(1,2)$ and $\mathbf{b}(2,1)$.

Under what condition on the change-of-basis matrix from one basis to another are both bases oriented in the same way? The question is posed both for the plane and for space.

What is the dimension of the vectors of the reciprocal basis $\mathbf{e}_{1}^{*}, \mathbf{e}_{2}^{*}, \mathbf{e}_{3}^{*}$ if the vectors of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ are measured in centimeters?

Suppose that the sides of the triangle $P_{1} P_{2} P_{3}$ are equal to the medians of the triangle $A_{1} A_{2} A_{3}$. It is required to find the ratio of the areas of these triangles.

Fig. 7